Round-six lemma `cplP_spec`

Proved

J6Seg.JK.cplP_spec

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

The following holds (Lean statement, in the scope of module Solutions.J6SegILJKer11):

theorem J6Seg.JK.cplP_spec {M : MulSpec} {Xb Φm Bx Bc : ℕ} (hMB : MulBnd M Xb Φm) (s : IState HS)
    {lp lp1 lh : List ℤ} (hF : CplF P F lp lp1) (hm : cc.length ≤ ze.length)
    (hwf : ∀ r (h : r < cc.length), cc[r].length + r ≤ ze.length)
    (hcB : ∀ r (h : r < cc.length), Bnd Xb cc[r]) (hzB : Bnd Xb ze) (hzl : ze.length < 2 ^ Bc)
    (hXB : 2 * Xb + P + Bc + 3 ≤ Bx) (hT : F dTJ = flatT cc ++ R) (hTT : F J6Seg.JK.TT = T0)
    (hE : F E = ze) (hW : F W = []) (hHM : F HM = (cc.length : ℤ) :: lh) :
    MSp M cplP (encK s F) (encK s (cSt P F cc ze R T0 cc.length))
      ((∑ r ∈ range cc.length, ((cc.getD r []).length + 3) + 1) * uC Φm Bx Bc)

Context. B-line IL, the J kernel (B4), part 11: the coupling window

Row r of cplP reads |cc_r|, runs the dot loop on the window E = ze.drop r and pushes W_r = dotF P cc_r (ze.drop r) onto W, then drops the head of E (cpl_body). After m = top HM rows W = (W_0, …, W_{m-1}).reverse (cplP_spec).

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_R6_Defs66

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.JK
open IntMul NLogN NLogN.KIL Function Finset
variable (P : ℕ) (F : Fin HS → List ℤ) (cc : List (List ℤ)) (ze R T0 : List ℤ)
variable {P F cc ze R T0}

Formal statement

theorem J6Seg.JK.cplP_spec {M : MulSpec} {Xb Φm Bx Bc : ℕ} (hMB : MulBnd M Xb Φm) (s : IState HS)
    {lp lp1 lh : List ℤ} (hF : CplF P F lp lp1) (hm : cc.length ≤ ze.length)
    (hwf : ∀ r (h : r < cc.length), cc[r].length + r ≤ ze.length)
    (hcB : ∀ r (h : r < cc.length), Bnd Xb cc[r]) (hzB : Bnd Xb ze) (hzl : ze.length < 2 ^ Bc)
    (hXB : 2 * Xb + P + Bc + 3 ≤ Bx) (hT : F dTJ = flatT cc ++ R) (hTT : F J6Seg.JK.TT = T0)
    (hE : F E = ze) (hW : F W = []) (hHM : F HM = (cc.length : ℤ) :: lh) :
    MSp M cplP (encK s F) (encK s (cSt P F cc ze R T0 cc.length))
      ((∑ r ∈ range cc.length, ((cc.getD r []).length + 3) + 1) * uC Φm Bx Bc) := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `J6Seg.JK.cplP_spec`, Solutions/J6SegILJKer11.lean (lines 146–194) of the round-six development.

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